Identifying Problematic Printers by Applying Markov Chain Model

ABSTRACT

Systems and methods for identifying problematic printers are provided. An example method can involve determining a time interval for a printing device. The method may also involve determining a number of pages printed by the printing device during the time interval. The method may also involve determining a number of printing-device errors that occur on the printing device during the time interval. Still further, the method may involve determining one or more coefficients of a Markov chain based on the number of pages printed by the printing device during the time interval and the number of printing-device errors that occur on the printing device during the time interval. The method yet further includes, based on the determined coefficients of the Markov chain, determining an operational status of the printing device.

BACKGROUND

Unless otherwise indicated herein, the description in this backgroundsection is not prior art to the claims in this application and is notadmitted to be prior art by inclusion in this section. The presentdisclosure relates to methods and systems for identifying problematicprinters by applying a Markov chain model.

Printing devices have increased in both number and complexity as aresult of added functions, applications, and services that can beprovided. As a result, the number and types of errors that occur onprinting devices have increased as well. Improving and/or simplifyingthe ability to identify problematic printers can lead to faster repairsand maintenance, and can lead to higher customer satisfaction and betterbusiness results.

SUMMARY

In a first aspect, an example method is provided. The method preferablyincludes determining a time interval for a printing device. The methodalso includes determining a number of pages printed by the printingdevice during the time interval. The method further includes determininga number of printing-device errors that occur on the printing deviceduring the time interval. The method still further includes determiningone or more coefficients of a Markov chain based on the number of pagesprinted by the printing device during the time interval and the numberof printing-device errors that occur on the printing device during thetime interval. The method yet further includes, based on the determinedcoefficients of the Markov chain, determining an operational status ofthe printing device.

In a second aspect, an example non-transitory computer readable mediumis provided. The non-transitory computer readable medium has storedthereon instructions that, when executed by a processor, causeperformance of a set of acts including determining a time interval for aprinting device. The set of acts also includes determining a number ofpages printed by the printing device during the time interval. The setof acts further includes determining a number of printing-device errorsthat occur on the printing device during the time interval. The set ofacts still further includes determining one or more coefficients of aMarkov chain based on the number of pages printed by the printing deviceduring the time interval and the number of printing-device errors thatoccur on the printing device during the time interval. The set of actsyet further includes, based on the determined coefficients of the Markovchain, determining an operational status of the printing device.

In a third aspect, a system is provided. The system preferably includesone or more processors. The system also includes a memory storage thathas instructions stored thereon that, when executed by the one or moreprocessors, cause the system to perform a set of acts. The set of actsincludes determining a time interval for a printing device. The set ofacts also includes determining a number of pages printed by the printingdevice during the time interval. The set of acts further includesdetermining a number of printing-device errors that occur on theprinting device during the time interval. The set of acts still furtherincludes determining one or more coefficients of a Markov chain based onthe number of pages printed by the printing device during the timeinterval and the number of printing-device errors that occur on theprinting device during the time interval. The set of acts yet furtherincludes, based on the determined coefficients of the Markov chain,determining an operational status of the printing device.

These, as well as other aspects, alternatives, and advantages, willbecome apparent to those of ordinary skill in the art by reading thefollowing detailed description, with reference where appropriate to theaccompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram illustrating an example system according to anembodiment of the present disclosure.

FIG. 2 is a block diagram illustrating an example computing deviceaccording to an embodiment of the present disclosure.

FIG. 3A is a diagram illustrating an example distribution ofprinting-device errors.

FIG. 3B is a diagram illustrating an example distribution ofprinting-device errors.

FIG. 4A is a diagram illustrating an example distribution of pagesprinted and printing-device errors over time.

FIG. 4B is a diagram illustrating an example distribution of pagesprinted and printing-device errors over time.

FIG. 4C is a diagram illustrating an example distribution of pagesprinted and printing-device errors over time.

FIG. 4D is a diagram illustrating example states according to an exampleembodiment of the present disclosure.

FIG. 5 is a block diagram illustrating an example Markov chain accordingto an embodiment of the present disclosure.

FIG. 6 is a block diagram illustrating an example Markov chain accordingto an embodiment of the present disclosure.

FIG. 7 is a flow chart illustrating an example method according to anembodiment of the present disclosure.

DETAILED DESCRIPTION

In the following detailed description, reference is made to theaccompanying figures, which form a part hereof. In the figures, similarsymbols typically identify similar components, unless context dictatesotherwise. The illustrative embodiments described in the detaileddescription, figures, and claims, are not meant to be limiting. Otherembodiments may be utilized, and other changes may be made, withoutdeparting from the scope of the subject matter presented herein. It willbe readily understood that the aspects of the present disclosure, asgenerally described herein, and illustrated in the figures, can bearranged, substituted, combined, separated, and designed in a widevariety of different configurations, all of which are explicitlycontemplated herein. Various functions described herein as beingperformed by one or more entities may be carried out by hardware,firmware, or software logic. For instance, various functions describedherein may be carried out be a processor executing instructions writtenin any suitable programming language and stored in memory.

In this description, the articles “a” or “an” are used to introduceelements of the example embodiments. The intent of using those articlesis that there is one or more of the elements. The intent of using theconjunction “or” within a described list of at least two terms is toindicate any of the listed terms or any combination of the listed terms.The use of ordinal numbers such as “first,” “second,” “third” and so onis to distinguish respective elements rather than to denote a particularorder of those elements.

I. OVERVIEW

Many organizations rely on properly functioning printing devices tooperate effectively. Schools, offices, courtrooms, hospitals, and othertypes of organizations may use one or more printers in order to printforms, contracts, and/or other documents needed for the organization torun. In the event an error occurs on a printing device, it may take timeand/or resources to fix the error and return the printing device to ahealthy state, which may adversely affect the organization. Some errors,such as an empty ink cartridge or paper tray, may be relatively simpleto fix. Other errors, such as a programming error or problem with aninternal component of the printing device, may be more difficult to fix.Further, some errors may be intermittent, not immediately apparent,and/or may not be easily diagnosed and fixed.

One of the most important metrics for a printing device is a ratio ofthe number of printed pages per error. For example, a printing devicethat has printed 10,000 pages in thirty days, while having a total of 50errors occur during the same duration, would have a ratio of:

$R = {\frac{10,000\mspace{14mu} {pages}}{50\mspace{14mu} {errors}} = {200\mspace{14mu} {pages}\mspace{14mu} {per}\mspace{14mu} {{error}.}}}$

This ratio on its own, however, only provides the average number ofpages printed per error for a given time interval, and does not includewhether the errors happened every day, whether there were 50 errors onone day, or some other distribution of errors. It also does not specifythe type of errors that occurred.

One purpose of the present disclosure is to provide methods and systemsfor identifying problematic printers. This may be done by determining anoperational status of a printing device using a Markov chain, andfiltering out printers that may seem problematic based on limitedinformation about their performance, but in reality are relativelyhealthy. In some cases, the operational status may be used to determinewhether maintenance personnel should be dispatched to the printingdevice, whether a printing device may be fixed remotely, or whether someother action should be taken.

First, a time interval is determined for a printing device. The timeinterval may define a duration for which the printing device is in adiscrete state of a Markov chain. At the end of each time interval, theprinting device may remain in its current discrete state, or maytransition to another discrete state of the Markov chain. For a givenprinting device, the time interval may be determined based on a contextin which the printing device is used, and may correspond to a frequencywith which a printing device is used and the need for reliable printingin the context. For instance, a time interval of twenty-four hours maybe determined for a printing device used in an office setting. On theother hand, where the context is a courtroom, a time interval of fourhours may be determined, because a courtroom may have a more frequentuse and higher need for reliable printing than an office, for example.Many other time intervals and factors used to determine a time intervalare possible as well.

Next, a number of pages printed by the printing device during the timeinterval may be determined. The number of pages printed may bedetermined based on a counter that increases each time a page isprinted, for example.

Further, a number of printing-device errors that occur on the printingdevice during the time interval may be determined. In some examples, aprinting device may keep track of printing-device errors by increasing acounter each time an error occurs. In other examples, eachprinting-device error may have an associated code or identifier whichmay allow the errors to be differentiated from each other.

An example method may then include determining one or more coefficientsof a Markov chain based on the number of pages printed by the printingdevice during the time interval and the number of printing-device errorsthat occur on the printing device during the time interval. A Markovchain of the present disclosure may include two states, such as ahealthy state and an unhealthy state. The determined coefficients mayinclude the transition probabilities associated with the probability oftransitioning from one state to another.

Then, based on the determined coefficients of the Markov chain, theexample method may include determining an operational status of theprinting device. Determining the operational status may includecomparing the determined coefficients to previous coefficients of thesame printing device and/or coefficients corresponding to other printingdevices. Example operational statuses may include a healthy status, anunhealthy status, or an unknown status. Other operational statuses arepossible as well.

II. EXAMPLE SYSTEMS AND PRINTING DEVICES

FIG. 1 is a block diagram illustrating an example system 100 accordingto an example embodiment of the present disclosure. System 100 mayinclude one or more printing devices 110A-110N, and a server 130, whichmay be connected via a network 120. In some examples, system 100 mayinclude more or fewer printing devices than are shown in FIG. 1, mayinclude additional servers and/or computing devices, and/or may includeone or more other systems or devices in addition to or instead of thoseshown in FIG. 1.

Printing devices 110A-110N may include one or more multi-functionprinting devices and/or stand-alone printing devices. A given printingdevice may be configured to perform one or more functions such asprinting, scanning, emailing, storing, modifying, receiving, ortransmitting one or more documents and/or files. In some examples,printing devices 110A-110N may include one or more computing devicessuch as computing device 200, or one or more components or aspects ofcomputing device 200 described in more detail with respect to FIG. 2. Insome examples, one or more of the printing devices 110A-110N may beconnected to one or more personal computers, laptops, servers, handhelddevices, and/or other computing devices and systems, which may be usedin connection with the printing device to perform one or more actions,such as those described above.

Each printing device 110A-110N may be configured to perform one or moresteps, actions, or functions described herein. For example, printingdevice 110A may communicate with server 130, to transmit and/or receivedata or information via network 120 including time intervals, numbers ofprinted pages, numbers of errors, and other related information.

Server 130 may include a cloud based server, for example, that canperform one or more tasks to manage and/or maintain printing devices110A-110N. Server 130 may communicate with printing devices 110A-110N totransmit or receive data. For instance, in some examples server 130 maytransmit a command to the one or more printing devices 110A-110N toreset, install updates, or perform one or more printing or maintenancefunctions or operations. In other examples, server 130 may receive datafrom the one or more printing devices 110A-110N, such as a page count(i.e., number of pages printed), an error count, one or more errormessages, or data corresponding to a page count, error count, and/orerror message.

In some examples, server 130 may be configured to perform one or morefunctions or steps of the example methods and systems disclosed herein.For instance, server 130 may determine a time interval, number of pagesprinted, and number of printing-device errors for one or more printingdevices. Server 130 may also determine a ratio of printed pages perprinting-device error for one or more printing devices over a given timeinterval. Further, server 130 may determine one or more Markov chaincoefficients based on the time interval, number of pages printed, numberof printing-device errors, and/or determined ratio. Server 130 may thendetermine an operational status of one or more printing devices, andtake one or more actions based on the determined operational status.

Server 130 may include one or more computing devices or systems (notshown), and may be consolidated in a single physical location, ordistributed across two or more physical locations. Server 130 mayinclude hardware, software, and/or firmware configured to carry out oneor more functions or acts described herein.

Network 120 in the system 100 may include one or more wired or wirelessconnections that support communication between the devices of system100. In some examples, network 120 may support one or more communicationprotocols, such as Extensible Messaging and Presence Protocol (XMPP),File Transfer Protocol (FTP), HyperText Transport Protocol (HTTP), JavaMessage Service (JMS), Simple Object Access Protocol (SOAP), ShortMessage Service (SMS), Simple Mail Transfer Protocol (SMTP), SimpleNetwork Management Protocol (SNMP), Transmission ControlProtocol/Internet Protocol (TCP/IP), User Datagram Protocol (UDP),Lightweight Directory Access Protocol (LDAP), and the Message Queue (MQ)family of network protocols.

Network 120 may be configured to allow communication between server 130and one or more printing devices 110A-110N, between the printing devices110A-110N themselves, and/or between one or more other devices orsystems and the system 100. Such communications may include commands,requests, and/or data corresponding to documents, printing-deviceerrors, and/or other data.

FIG. 2 is a block diagram illustrating an example computing device 200according to an embodiment of the present disclosure. In someembodiments, computing device 200 may be configured for performing avariety of functions or acts, such as those described in this disclosurewith respect to printing devices 110A-110N and server 130 of system 100,and with respect to one or more methods and systems described herein.

Computing device 200 may include one or more components, including forexample, one or more processors 210, communication interfaces 220, userinterfaces 230, and data storages 240. The components of computingdevice 200 may be communicatively connected to each other (or otherdevices or systems) via a system bus, network, or other link 250.

Processor 210 in computing device 200 may include one or more generalpurpose processors, central processing units (CPUs), CPU cores, and/orone or more special purpose processors (e.g., graphics processing units(GPUs), digital signal processors (DSPs), field programmable gatedarrays (FPGAs), application specific integrated circuits (ASICs), etc.).Processor 210 may be configured to execute computer-readable programinstructions contained in data storage 240, and/or other instructions asdescribed herein.

Communication interface 220 in computing device 200 may be configured toallow computing device 200 to communicate with one or more devices (orsystems) according to one or more protocols. In one example,communication interface 220 may be a wired interface, such as anEthernet interface or a USB interface. As another example, communicationinterface 220 may be a wireless interface, such as a cellular or Wi-Fiinterface, for example. Other example communication interfaces mayinclude an Ethernet connection, Universal Serial Bus (USB) connection,fiber-optic link, coaxial cable, Bluetooth, ZigBee, WiMAX, wirelesswide-area network (WWAN), and/or other similar type of interface.

Communication interface 220 of computing device 200 may allow one ormore commands, requests, and/or data to be transmitted between thecomputing device and one or more other computing devices or systems,such as between one of printing devices 110A-110N and server 130 insystem 100.

User interface 230 in computing device 200 may facilitate interactionwith a user of the computing device, if applicable. As such, userinterface 230 may include input components such as a keyboard, keypad,computer mouse, trackball, joystick, camera, microphone, voicerecognition module, and touch sensitive panel, and output componentssuch as a display screen (which, for example, may be combined with atouch sensitive panel), sound speaker, and haptic feedback system.

Data storage 240 in computing device 200 may include one or morevolatile, non-volatile, removable, and/or non-removable storagecomponents, such as magnetic, optical, or flash storage, and may beintegrated in whole or in part with processor 210. Data storage 240 maystore machine-readable instructions that, when executed by processor210, cause performance of a set of acts or functions such as thosedescribed herein with respect to the devices, systems, methods,features, and embodiments. Such program instructions may define or bepart of a discrete software application that can be executed in responseto certain inputs received from user interface 230, for instance. Datastorage 240 may also store other types of information or data, such asthose types described throughout this disclosure (e.g., time interval,number of pages printed, number of printing-device errors, error typesor error codes, and/or other data).

III. EXAMPLE ERROR DISTRIBUTIONS

FIGS. 3A and 3B show diagrams illustrating example distributions ofprinting-device errors according to embodiments of the presentdisclosure. FIG. 3A shows a diagram corresponding to a printing devicewhich has errors spread evenly over a given number of printed pages.Axis 310 corresponds to a number of printed pages and axis 320corresponds to the type of error, while each mark 330 corresponds to anerror. The types of errors that are distributed evenly may be systemicerrors, errors that occur due to the age of the printing device, orother types of errors that are difficult to easily diagnose and fix. Forexample, a programming bug may cause an attempt to print certain typesof documents to cause a printing-device error, which may not be easilynoticeable. Further, as the printing device ages, one or more componentsmay lose their ability to function properly, which may lead to errorsoccurring. On the other hand, FIG. 3B shows a diagram corresponding to aprinting device which has errors concentrated over two shorter amountsof printed pages. Axis 310 again corresponds to a number of printedpages and axis 320 corresponds to the type of error, while each mark 330corresponds to an error. In general, errors that occur in rapidsuccession and are fixed quickly are the types of errors that are easilyfixed, such as a paper jam or empty paper tray. While both FIGS. 3A and3B show the same number of errors over the same number of printed pages(40/15,000), the operational status of the printing devices may bedifferent, where one is unhealthy and one is healthy for example.

In FIG. 3A, the horizontal axis 310 may represent the number of pagesprinted by a given printing device over time, such as one of printingdevice 110A-110N. The vertical axis 320 may represent the type of errorthat occurred on the printing device. Each mark 330 may represent aprinting-device error that occurred on the printing device. In theexample shown in FIG. 3A, the corresponding printing device has printed15,000 pages, and has had 40 errors occur. The errors were generallyevenly spaced throughout the 15,000 printed pages. The ratio of pagesprinted per error in FIG. 3A is:

${R\left( {3A} \right)} = {\frac{15,000\mspace{14mu} {pages}}{40\mspace{14mu} {errors}} = {375\mspace{14mu} {pages}\mspace{14mu} {per}\mspace{14mu} {{error}.}}}$

In FIG. 3B, the horizontal axis 310 and vertical axis 320 represent thenumber of pages printed and type of error that occurred on a givenprinting device respectively, similar to FIG. 3A. FIG. 3B differs fromFIG. 3A, however, in that the distribution of errors is moreconcentrated over the number of printed pages. For instance, FIG. 3Bshows that there were 40 errors in total, with 20 errors having occurredbetween the printing of page 0 and page 5000, while 20 occurred betweenthe printing of page 10,000 and page 15,000. However, the ratio ofprinted pages per error for the printing device corresponding to FIG. 3Bis:

${R\left( {3B} \right)} = {\frac{15,000\mspace{14mu} {pages}}{40\mspace{14mu} {errors}} = {375\mspace{14mu} {pages}\mspace{14mu} {per}\mspace{14mu} {{error}.}}}$

Even though the distribution of errors is different for the printingdevices corresponding to the diagrams in FIGS. 3A and 3B, the ratio oferrors is the same. As such, it is difficult to differentiate printingdevices based on the ratio alone.

FIGS. 4A and 4B show diagrams illustrating example distributions ofprinting-device errors according to embodiments of the presentdisclosure. FIG. 4A shows a diagram corresponding to a printing devicewhich has errors spread evenly over a given duration of time. Theprinting device corresponding to FIG. 4A may be similar or identical insome respects to the device corresponding to FIG. 3A. For instance, theerrors may be evenly distributed over time, as shown in FIG. 4A. Theprinting device corresponding to FIG. 4B may be similar or identical insome respects to the printing device corresponding to FIG. 3B. Forinstance, the errors may be concentrated over a shorter duration of timeor fewer number of printed pages. While both figures show the sameamount of errors over the same number of printed pages, the status ofthe printing devices may be different, where one is unhealthy and one ishealthy for example.

In FIG. 4A, the horizontal axis 410 may be a time axis, where T0 is afirst point in time, T1 is a second point in time, and T2 is a thirdpoint in time. Vertical axis 420 may represent the number of pagesprinted by a corresponding printing device, or a value of a counter ofthe printing device. Vertical axis 430 may represent the types ofprinting-device errors that occur on the corresponding printing device,such as a paper jam, network error, or others. Each mark 440 mayrepresent a printing-device error. In the example shown in FIG. 4A,between T0 and T2, the corresponding printing device printed 2,000 pagesand had six errors occur. The errors were generally evenly spacedbetween T0 and T2. The ratio of pages printed per error in FIG. 4A forthe full time interval shown in FIG. 4A, (i.e., the time intervalbetween T0 and T2) is:

${R\left( {{4A},{{T\; 0} - {T\; 2}}} \right)} = {\frac{2,000\mspace{14mu} {pages}}{6\mspace{14mu} {errors}} = {{\sim 333}\mspace{14mu} {pages}\mspace{14mu} {per}\mspace{14mu} {{error}.}}}$

The ratios of printed pages per error for the time intervals of T0 to T1and T1 to T2 respectively are:

${R\left( {{4A},{{T\; 0} - {T\; 1}}} \right)} = {\frac{1,000\mspace{14mu} {pages}}{3\mspace{14mu} {errors}} = {{{\sim 333}\mspace{14mu} {pages}\mspace{14mu} {per}\mspace{14mu} {{error}.{R\left( {{4A},{{T\; 1} - {T\; 2}}} \right)}}} = {\frac{1,000\mspace{14mu} {pages}}{3\mspace{14mu} {errors}} = {{\sim 333}\mspace{14mu} {pages}\mspace{14mu} {per}\mspace{14mu} {{error}.}}}}}$

In FIG. 4B, the horizontal axis 410 and vertical axes 420 and 430 mayrepresent time, the number of pages printed (or a counter value), andtype of printing-device errors respectively for a given printing device,similar to the axes of FIG. 4A. FIG. 4B differs from FIG. 4A, however,in that the distribution of errors 450 is more concentrated over time.For instance, FIG. 4B shows that there were 6 errors total, with fourerrors occurring between T0 and T1 and 2 errors occurring between T1 andT2. The ratio of printed pages per error in FIG. 4B, for the full timeinterval between T0 and T2 is:

${R\left( {{4B},{{T\; 0} - {T\; 2}}} \right)} = {\frac{2,000\mspace{14mu} {pages}}{6\mspace{14mu} {errors}} = {{\sim 333}\mspace{14mu} {pages}\mspace{14mu} {per}\mspace{14mu} {{error}.}}}$

The ratios of printed pages per error for the time intervals of T0 to T1and T1 to T2 respectively are:

${R\left( {{4B},{{T\; 0} - {T\; 1}}} \right)} = {\frac{1,000\mspace{14mu} {pages}}{4\mspace{14mu} {errors}} = {{250\mspace{14mu} {pages}\mspace{14mu} {per}\mspace{14mu} {{error}.{R\left( {{4B},{{T\; 1} - {T\; 2}}} \right)}}} = {\frac{1,000\mspace{14mu} {pages}}{2\mspace{14mu} {errors}} = {500\mspace{14mu} {pages}\mspace{14mu} {per}\mspace{14mu} {{error}.}}}}}$

As can be seen from the ratios corresponding to FIGS. 4A and 4B, theselection of time intervals (T0, T1, and T2) has a noticeable impact onthe calculated ratios. These ratios may be used in part to determinecoefficients of one or more Markov chains, which may then be used todetermine an operational status of a printing device.

FIG. 4C shows a diagram illustrating an example distribution ofprinting-device errors according to an embodiment of the presentdisclosure. In the embodiment shown in FIG. 4C, the state of theprinting device is determined based on (i) the type of errors that occurduring a time interval and (ii) the number of errors that occur duringthe same time interval. Axis 410 corresponds to the time, axis 420corresponds to the number of printed pages, and axis 430 corresponds tothe type of errors that occur. Each mark 450 corresponds to an error. Inthis example, errors may be categorized as “low cost” errors (i.e.,errors that cause a short down time of the printing device, such as lowtoner), and “high cost” errors (i.e., errors that require a long downtime of the printing device, such as a system error). Other categoriesare possible as well, such as “medium cost” or others. Line 460 in FIG.4C illustrates a line distinguishing between high cost and low costerrors. In examples that include “low cost,” “high cost,” and “mediumcost” errors (and/or other categories), there may be two (or more) linesseparating the respective categories.

In the example shown in FIG. 4C, there is a threshold value for thenumber of errors that occur during a time interval. Here, the thresholdmay be set at four errors. In other examples, there may be multiplethresholds, such as four errors, fifty errors, and one hundred errors,such that the number of errors that occur may fall into a singlecategory that is either zero-to-four errors, five-to-fifty errors,fifty-to-one hundred errors, or greater than one hundred errors. Othervalues are possible as well.

In FIG. 4C, the example printing device has printed 2,000 pages betweenT0 and T2. Between T0 and T1, one low cost error occurred. Between T1and T2, 5 high cost errors occurred. Using a threshold number of errorsset at four errors, between T0 and T1, the printing device was “belowthreshold” and “low cost,” and between T1 and T2, the printing devicewas “above threshold” and “high cost.”

FIG. 4D is a diagram 480 showing the possible states for the exampleprinting device having an error distribution according to FIG. 4C. State1 (S1) is a state in which fewer than the threshold number of “highcost” errors (e.g., four or less) occur during a time interval. State 2(S2) is a state in which greater than the threshold number of “highcost” errors occur during a time interval. State 3 (S3) is a state inwhich fewer than the threshold number of “low cost” errors occur duringa time interval. And state 4 (S4) is a state in which greater than thethreshold number of “low cost” errors occur during a time interval.

For the example shown in FIGS. 4C and 4D, the printing device is instate S3 for the first time interval, because only one low cost erroroccurred. The printing device then transitions to state S2 for thesecond time interval, because five high cost errors occurred. Putanother way, the state of the printing device in this example isdetermined based on both the number and type of errors that occur duringeach time interval. At the end of each interval, the printing device mayor may not transition to a different state, depending on the number andtype of errors that occurred during the previous time interval.

IV. EXAMPLE MARKOV CHAINS

A Markov chain describes a process that undergoes transitions from onestate to another on a state space. The one or more states may correspondto states of a system or device, such as system 100 or one or more ofprinting devices 110A-110N. Each Markov chain may have one or moreassociated coefficients, which are used to determine a probability oftransitioning between states or remaining in a given state of the chain.For instance, a two-state Markov chain, such as the one shown in FIG. 5,may have a probability of transitioning from a first state to a secondstate, from the second state to the first state, and a probability ofremaining in each state, for a total of four coefficients or transitionprobabilities.

FIG. 5 is a block diagram illustrating an example Markov chain accordingto an embodiment of the present disclosure. The Markov chain in FIG. 5is a two-state Markov chain including a healthy state 501 and anunhealthy state 502, which correspond to possible states of a printingdevice, such as one of printing devices 110A-110N. Each state may bedefined by one or more variables. For example, healthy state 501 andunhealthy state 502 may be defined as:

-   -   State 501(Healthy)=R>F1    -   State 502(Unhealthy)=R<F2 where R is the ratio of the number of        pages printed per printing-device error over a given time        interval, and F1 and F2 are threshold values. The threshold        values may be set in advance, or may be dynamically changed        based on data received from one or more devices or systems.

In one particular example, threshold value F1 may be 2000 pages pererror, and threshold F2 may be 100 pages per error for a given Markovchain. For each time interval, a ratio is determined. Suppose that agiven printing device, such as printing device 110A, is in healthy state501 at a first time T0. At time T1 after a determined time interval haselapsed, a ratio of printed pages per error is determined (i.e.,R(T0−T1)). If the determined ratio is less than 100 pages per error(i.e., R<F2), then the printing device transitions to unhealthy state502. However, if the ratio is greater than or equal to 100, the printingdevice remains in healthy state 501.

Now suppose that printing device 110A is in unhealthy state 502 at timeT1. At time T2 after a determined time interval has elapsed, the ratioof printed pages per error is again determined (i.e., R(T1−T2)). If thedetermined ratio is greater than 2000 pages per error (i.e., R>F1), theprinting device may transition back to healthy state 501. However, ifthe ratio is less than or equal to 2000 pages per error, the printingdevice may remain in unhealthy state 502.

FIG. 5 includes transition paths 511, 512, 521, and 522, correspondingto the probability of remaining in state 501 (511), transitioning fromstate 501 to state 502 (512), transitioning from state 502 to state 501(521), and remaining in state 502 (522). Each of these probabilities maycorrespond to, include, or be based on one or more coefficients, whichin turn may be determined in part by one or more threshold valuesdetermined for a given printing device.

FIG. 6 shows a block diagram illustrating another example Markov chainaccording to an embodiment of the present disclosure. FIG. 6 shows athree-state Markov chain, where the states are a healthy state 601, anunhealthy state 602, and an unknown state 603. Unknown state 603 maycorrespond to a state of a printing device that is not in use or has avery low usage (e.g., a small number of printed pages or small durationof time in use). The Markov chain may also have transition paths 611,612, 613, 621, 622, 623, 631, 632, and 633, which correspond to theprobability of a printing device remaining in a given state, ortransitioning to a different state, from one time interval to the next.The transition paths in FIG. 6 may correspond to the probability ofremaining in healthy state 601 (611), transitioning from healthy state601 to unhealthy state 602 (612), transitioning from healthy state 601to unknown state 603 (613), transitioning from unhealthy state 602 tohealthy state 601 (621), remaining in unhealthy state 602 (622),transitioning from unhealthy state 602 to unknown state 603 (623),transitioning from unknown state 603 to healthy state 601 (631),transitioning from unknown state 603 to unhealthy state 602 (632), andremaining in unknown state 603 (633).

In one example, states 601, 602, and 603 may be defined as

-   -   State 601(Healthy)=R>F1    -   State 602(Unhealthy)=R<F2    -   State 603(Unknown)=PP<1000

where R is the ratio of the number of pages printed per printing-deviceerror over a given time interval, F1 and F2 are threshold values, and PPis the number of pages printed during a given time interval.

Following the example as described with respect to FIG. 5, thresholdvalue F1 may be 2000 pages per error and threshold F2 may be 100 pagesper error for a given Markov chain. In addition, the printed pages (PP)value may be 1000 pages. For each time interval, a ratio (R) isdetermined. Suppose that printing device 110A is in healthy state 601 ata first time T0. At time T1 after a determined time interval haselapsed, a ratio of printed pages per error is determined (i.e.,R(T0−T1)). If the determined ratio is less than 100 pages per error(i.e., R<F2), then printing device 110A transitions to unhealthy state602. However, if the ratio is greater than or equal to 100, printingdevice 110A remains in healthy state 601.

Now suppose that printing device 110A is in unhealthy state 602 at timeT1. At time T2 after a determined time interval has elapsed, the ratioof printed pages per error is again determined (i.e., R(T1−T2)). If thedetermined ratio is greater than 2000 pages per error (i.e., R>F1),printing device 110A may transition back to healthy state 601. However,if the ratio is less than or equal to 2000 pages per error, printingdevice 110A may remain in unhealthy state 602.

Suppose further, that printing device 110A is in either healthy state601 or unhealthy state 602 at time T2. At time T3 after a determinedtime interval has elapsed, the ratio of printed pages per error is againdetermined. (As described above, determining the ratio includesdetermining the number of pages printed and dividing by the number ofprinting-device errors.) If the number of pages printed during the timeinterval from T2-T3 is less than 1000 pages, printing device 110A willtransition to unknown state 603. In some examples, the number of pagesprinted is prioritized over a determined ratio. For instance, whereprinting device 110A has printed 50 pages during a time interval and hashad 5 errors occur (i.e., R=10 and PP=50), printing device 110A may haveconflicting instructions to (i) transition to unhealthy state 602(because R<100), and (ii) transition to unknown state 603 (becausePP<1000). The number of pages printed may be prioritized, such that inthis example printing device 110A transitions to or remains in unknownstate 603. It should be noted that the thresholds, values, andpriorities in these examples are for illustrative purposes only, and maybe set and modified based on a desired implementation.

While FIGS. 5 and 6 are shown with states corresponding to healthy,unhealthy, and unknown, it should be noted that other states arepossible as well. For example, various states may be determined based onthe number and type of errors that occur (i.e., high cost and belowthreshold, high cost and above threshold, etc.) as described withrespect to figured 4C and 4D above.

V. EXAMPLE METHODS

FIG. 7 is a flow chart illustrating an example method 700 according toan embodiment of the present disclosure. Although FIG. 7 illustratesmethod 700 as including certain blocks in a particular order, it shouldbe understood that blocks may be added, subtracted, and/or carried outin a different order while remaining within the scope of thisdisclosure. Furthermore, some or all of the blocks of method 700 may becarried out by system 100 and/or the various components of system 100,including printing devices 110A-110N, for example.

Method 700 may include blocks 702-710. At block 702, method 700 mayinvolve determining a time interval for a printing device. The printingdevice may be one of printing devices 110A-110N in FIG. 1, for example.The time interval may be any duration of time over which datacorresponding to the printing device can be gathered. For instance, thetime interval may be twenty-four hours, one week, or longer, or may beas short as an hour or less, depending on the frequency and type of use,as well as other factors. The time interval may be determined such thatit allows the printing device time to print a sufficient number ofpages. For instance, for a printing device that is used often and printsa substantial amount of pages during normal operation, a shorter timeinterval may be determined. However, for a printing device that is notused very often, for example a few times a day, a longer time intervalmay be determined, such as a week or longer.

In some examples, the time interval for a given printing device may bedetermined based on a context in which the printing device operates. Thecontext may be a business, office, shop, hospital, school, courtroom,law office, home, or any other context in which a printing deviceoperates. The context may have one of more associated factors used todetermine the time interval for the context, such as how often theprinting device is used in the context, the usual type of documentprinted, the type of use of the printing device (scanning, emailing,copying, printing, etc. . . . ), the likelihood that the printing devicewill have an error occur during a given time interval, the importance ofhaving a functioning printing device in the context, or other factors.

In some examples, the time interval for a printing device may bedetermined by the printing device itself, by a server connected to theprinting device, or by another device or system connected to theprinting device and/or server. After the time interval is determined, insome cases the time interval may be transmitted from the server to theprinting device, or vice versa, or from another device or system to theprinting device and/or server. As a result, the time interval may bedetermined by one device in the system, and may be used by anotherdevice in the system to perform one or more functions or acts describedherein.

In some examples, the time interval for a printing device may bedetermined independently from one or more other printing devices. Forinstance, each printing device 110A-110N in system 100 may have a timeinterval determined independently from the other printing devices. Theserver 130 may determine a time interval for each printing device110A-110N based on the context of each printing device, and/or otherfactors or information. Server 130 may receive, or have stored in amemory, information corresponding to each printing device 110A-110N. Forexample, server 130 may store information corresponding to the contextin which each printing device 110A-110N is used, the type and age ofeach printing device, and/or statistical information on how eachprinting device is used. Server 130 may determine a time interval foreach printing device based on the corresponding information. In thatway, where two or more printing devices 110A-110N are connected toserver 130, a different time interval may be determined for eachprinting device.

In other examples, the time interval for a printing device in a systemincluding two or more printing devices may be determined based on datacorresponding to one or more other printing devices. For instance, asystem including two or more printing devices may determine a singletime interval to be used for both printing devices, or may use datacorresponding to one printing device to determine a time interval forthe second device. In a system including many printing devices, data maybe gathered corresponding to time intervals, pages printed, printingdevice errors, and operational statuses of the many printing devices,which may then be used to determine a time interval for a printingdevice in the system. For instance, a time interval for one printingdevice may be determined based on one or more time intervals determinedfor similar printing devices in the system, such as printing devices ina similar context, similar age, similar pages printed and errorsoccurred, similar types of errors, and/or one or more othercharacteristics.

At block 704, method 700 may involve determining a number of pagesprinted by the printing device during the time interval. In someexamples, the printing device may include a counter that counts thenumber of pages printed by the printing device. The counter mayincrement by one each time a page is printed. In other examples, theprinting device may associate a time stamp with each printed page ordocument. The printing device may determine a number of printed pagesduring the time interval based on the counter, the time stamp, and/orother information.

In some examples, the printing device may transmit data to a server,such as server 130 in system 100. The data may include a page count fromthe counter of the printing device, a number of pages printed during thetime interval, or any other relevant information. In a particularexample, at a time T0 printing device 110A may transmit a first countervalue to server 130. At T1, after a time interval has elapsed, printingdevice 110A may transmit a second counter value to server 130. Server130 may then subtract the first counter value from the second countervalue to determine a number of pages printed between T0 and T1. In otherexamples, printing device 110A may perform a subtraction of the countervalues, and then provide the result to server 130. Other examples arepossible as well.

At block 706, method 700 may involve determining a number ofprinting-device errors that occur on the printing device during the timeinterval. The errors may include errors that are easily fixed, such asan empty paper tray or paper jam, as well as errors that are not easilyfixed, such as communication errors, broken components, or other complexproblems. Each error that occurs on the printing device may have acorresponding error code that may be used to identify the type of error,the time the error occurred, and other error details.

In some examples, the printing device may include a counter thatincrements for each error that occurs on the printing device. Theprinting device may then transmit the counter value to a server such asserver 130 in system 100. In other examples, the printing device maytransmit an error code to the server each time an error occurs. In stillother examples, the printing device may store information correspondingto each error that occurs, and transmit the information to the server ata specified interval, such as once per day. In yet other examples, theprinting device may store information corresponding to the number and/ortype of error that occurred on the printing device, and may transmitthat information to the server at the end of the time intervaldetermined for the printing device.

At block 708, method 700 may involve determining one or morecoefficients of a Markov chain. The Markov chain may be similar oridentical to the Markov chains described above with respect to FIGS. 5and 6. For instance, the Markov chain may include two or more statescorresponding to an operational status of a printing device.

In some examples, the determined coefficients may be based on the numberof pages printed by the printing device during the time interval and thenumber of printing device errors that occur on the printing deviceduring the time interval. For instance, the coefficients of the Markovchain may be the probability of transitioning from one state to another,and may be determined based on the printed pages and/or errors for agiven printing device. As described above with respect to FIG. 6,unknown state 603 may correspond to a printing device that has printedless than 1000 pages during a time interval. As such, the coefficientscorresponding to transition paths 613 and 623 for a printing devicetransitioning from healthy state 601 or unhealthy state 602 to unknownstate 603 may be determined based on the number of pages printed by theprinting device, and not based on the number of errors that occurred onthe printing device during the same time interval.

In some examples, the number of pages printed during a time interval maybe divided by the number of printing-device errors that occur on theprinting device during the time interval, resulting in a ratio ofprinted pages per error. The ratio may then be used to determine thecoefficients (i.e., transition probabilities) of a Markov chain. Forinstance, as described above with respect to FIG. 6, healthy state 601may correspond to a printing device that has a ratio lager than 2000pages per error, and unhealthy state 602 may correspond to a printingdevice having a ratio less than 100 pages per error. The coefficientscorresponding to a transition from one state into the healthy orunhealthy state (i.e., paths 612, 621, 632, and 632) may thus bedetermined based on the ratio determined for the printing device.

At block 710, method 700 may involve, based on the determined one ormore coefficients of the Markov chain, determining an operational statusof the printing device. Determining the operational status may includecomparing the determined coefficients to previous coefficients of thesame printing device. Current and previous coefficients may be stored,and a comparison between past coefficients and current coefficients mayallow the printing device to be ranked and/or classified. The comparisonmay also allow the printing device, server, and/or another system ordevice to determine more precisely which errors or types of errorsoccur, and whether any components need to be replaced.

Determining the operational status may also include comparing thedetermined coefficients to coefficients of other printing devices. Thecomparison between printing devices may allow the printing devices to beranked and/or compared to an average health level or threshold healthlevel, and may allow easier determination of which device or devicesrequire maintenance.

As discussed above, the operational status may be a healthy status, anunhealthy status, or an unknown status, for example. Other statuses maybe possible as well such as a maintenance status, which may indicate aprinting device currently undergoing maintenance, or another status,which may be a catch-all or default status.

In some examples, a server or other device or system connected to aprinting device may classify the printing device as healthy, unhealthy,or unknown (or another status) based on the Markov chain, including theMarkov chain coefficients. For instance, the server, device, or systemmay build the Markov chain using the time interval, number of printedpages during the time interval, and number of printing-device errorsthat occur during the time interval. The server, device, or other systemmay include software, or computer program instructions encoded on acomputer-readable storage medium in a machine-readable format. Thecomputer program instructions, when executed by a processor, mayimplement one or more actions such as receiving data from one or moredevices or systems, building a Markov chain, and classifying printingdevices based on the Markov chain, for example. The computer programinstructions can implement one or more actions described in thisdisclosure by determining and/or carrying out one or more equations orformulas, comparing data from two or more printing devices, and/orperforming other logical operations.

In some examples, the server, device, or system may determine the statusof a printing device based on a most recent determined ratio, or basedon one or more other factors. In other examples, the server maydetermine the status of a printing device based on an analysis ofmultiple printing devices and/or multiple Markov chains. The server maybe a central server connected to a plurality of printing devices, andmay use information received from one or more other printing devices todetermine the operational status of a given printing device.

In still other examples, after or responsive to determining theoperational status of a printing device, the server, device or systemmay transmit a message to the printing device indicating the status ofthe printing device. The server, device, or system may also transmit oneor more commands, data, or information.

VI. EXAMPLE VARIATIONS

In some embodiments, the printing device may be a first printing deviceof a plurality of printing devices. For instance, as shown in FIG. 1,system 100 includes a plurality of printing devices 110A-110N. Themethods and operations described in this disclosure may be performed forthe plurality of printing devices 110A-110N.

For example, where a system includes a plurality of printing devices110A-110N, coefficients for a shared Markov chain may be determinedbased on the time intervals, pages printed, and printing device errorsof all or a subset of the printing devices 110A-110N. Further, theoperational status of each of the plurality of printing devices110A-110N may be determined based on the coefficients of the sharedMarkov chain.

In other examples, coefficients for a plurality of Markov chains may bedetermined for a plurality of printing devices. For instance, eachprinting device 110A-110N of system 100 may have an associated Markovchain and coefficients, which may be used to determine the operationalstatus of each printing device 110A-110N.

In still other examples, one or more coefficients of a Markov chaincorresponding to a first printing device may be used to determine thecoefficients of a second Markov chain corresponding to a second printingdevice.

VII. CONCLUSION

The present disclosure is not to be limited in terms of the particularembodiments described in this application, which are intended asillustrations of various aspects. Many modifications and variations canbe made without departing from its spirit and scope, as will be apparentto those skilled in the art. Functionally equivalent methods andapparatuses within the scope of the disclosure, in addition to thoseenumerated herein, will be apparent from the foregoing descriptions.Such modifications and variations are intended to fall within the scopeof the appended claims.

With respect to any or all of the diagrams, examples, and flow charts inthe figures and as discussed herein, each block and/or connection mayrepresent a processing of information and/or a transmission ofinformation in accordance with example embodiments. Alternativeembodiments may be included within the scope of such exampleembodiments. Further, more or fewer blocks and/or functions may be usedwith any of the diagrams, examples, and flow charts discussed herein,and these diagrams, examples, and flow charts may be combined with oneanother, in part or in whole.

A step or block that represents a processing of information maycorrespond to circuitry that can be configured to perform the specificlogical functions of a herein-described method or technique.Alternatively or additionally, a step or block that represents aprocessing of information may correspond to a module, a segment, or aportion of program code (including related data). The program code mayinclude one or more instructions executable by a processor forimplementing specific logical functions or actions in the method ortechnique. The program code and/or related data may be stored on anytype of computer-readable medium, such as a storage device, including adisk drive, a hard drive, or other storage media.

The computer-readable medium may also include non-transitorycomputer-readable media such as computer-readable media that stores datafor short periods of time like register memory, processor cache, and/orrandom access memory (RAM). The computer-readable media may also includenon-transitory computer-readable media that stores program code and/ordata for longer periods of time, such as secondary or persistent longterm storage, like read only memory (ROM), optical or magnetic disks,and/or compact-disc read only memory (CD-ROM), for example. Thecomputer-readable media may also be any other volatile or non-volatilestorage systems. A computer-readable medium may be considered acomputer-readable storage medium, for example, and/or a tangible storagedevice.

Additionally, any enumeration of elements, blocks, or steps in thisspecification, the drawings, or the claims is for purposes of clarity.Thus, such enumeration should not be interpreted to require or implythat these elements, blocks, or steps adhere to a particular arrangementor are carried out in a particular order.

It should be understood that arrangements described herein are forpurposes of example only. As such, those skilled in the art willappreciate that other arrangements and other elements (e.g., machines,interfaces, functions, orders, groupings of functions, etc.) can be usedinstead, and some elements may be omitted altogether according to thedesired results. Further, many of the elements that are described arefunctional entities that may be implemented as discrete or distributedcomponents or in conjunction with other components, in any suitablecombination and location.

While various aspects and embodiments have been disclosed herein, otheraspects and embodiments will be apparent to those skilled in the art.The various aspects and embodiments disclosed herein are for purposes ofillustration and are not intended to be limiting with the true scope andspirit being indicated by the following claims.

1. A method comprising: determining a plurality of time intervals for aprinting device including a first time interval and a second timeinterval; determining a first number of pages printed by the printingdevice during the first time interval and a second number of pagesprinted by the printing device during the second time interval, whereinthe first and second number of pages printed are determined by a counterthat is incrementally increased for each page printed by the printingdevice; determining a first number of printing-device errors that occuron the printing device during the first time interval and a secondnumber of printing-device errors that occur on the printing deviceduring the second time interval, wherein each printing-device error hasa corresponding error type; calculating a first ratio of pages printedper error, based on the first number of pages and the first number ofprinting-device errors, and calculating a second ratio of pages printedper error, based on the second number of pages and the second number ofprinting-device errors; comparing the first ratio to one or morepredetermined thresholds, and comparing the second ratio to the one ormore predetermined thresholds; determining, based on the comparison ofthe first ratio with the one or more predetermined thresholds, a firstoperational state of the printing device during the first time interval,wherein the first operational state is one of two or more operationalstates; determining, based on the comparison of the second ratio withthe one or more predetermined thresholds, a second operational state ofthe printing device during the second time interval, wherein the secondoperational state is different from the first operational state and isone of the two or more operational states; determining a set of one ormore coefficients of a Markov chain based on the first and secondoperational states, and one or more transitions between the two or moreoperational states; and based on the determined set of coefficients,determining an operational status of the printing device.
 2. The methodof claim 1, wherein the first and second time intervals for the printingdevice are determined based on a context in which the printing deviceoperates.
 3. The method of claim 2, wherein the context comprises abusiness office, and wherein the time interval for the printing deviceis a business day.
 4. (canceled)
 5. The method of claim 1, whereindetermining the set of one or more coefficients of the Markov chainbased on the first and second operational states, and the transitionsbetween the two or more operational states comprises: determining theset of one or more coefficients of the Markov chain based on thetransition from the first operational state to the second operationalstate.
 6. The method of claim 1, further comprising: based on thedetermined operational status of the printing device, classifying theprinting device as either healthy, unhealthy, or unknown.
 7. The methodof claim 1, further comprising: based on the determined operation statusof the printing device, transmitting a message indicating the status ofthe printing device.
 8. The method of claim 1, wherein the printingdevice is a first printing device of a plurality of printing devices,and wherein determining the operational status of the printing devicebased on the determined set of coefficients of the Markov chain furthercomprises: determining the operational status of each of the pluralityof printing devices based on the determined set of coefficients of theMarkov chain.
 9. A non-transitory, computer-readable medium, havinginstructions stored thereon, that, when executed by one or moreprocessors, cause performance of a set of acts comprising: determining aplurality of time intervals for a printing device including a first timeinterval and a second time interval; determining a first number of pagesprinted by the printing device during the first time interval and asecond number of pages printed by the printing device during the secondtime interval; determining a first number of printing-device errors thatoccur on the printing device during the first time interval and a secondnumber of printing-device errors that occur on the printing deviceduring the second time interval, wherein each printing-device error hasa corresponding error type; calculating a first ratio of pages printedper error, based on the first number of pages and the first number ofprinting-device errors, and calculating a second ratio of pages printedper error, based on the second number of pages and the second number ofprinting-device errors; comparing the first ratio to one or morepredetermined thresholds, and comparing the second ratio to the one ormore predetermined thresholds; determining, based on the comparison ofthe first ratio with the one or more predetermined thresholds, a firstoperational state of the printing device during the first time interval,wherein the first operational state is one of two or more operationalstates; determining, based on the comparison of the second ratio withthe one or more predetermined thresholds, a second operational state ofthe printing device during the second time interval, wherein the secondoperational state is different from the first operational state and isone of the two or more operational states; determining a set of one ormore coefficients of a Markov chain based on the first and secondoperational states, and one or more transitions between the two or moreoperational states; and based on the determined set of coefficients,determining an operational status of the printing device.
 10. (canceled)11. (canceled)
 12. (canceled)
 13. (canceled)
 14. (canceled)
 15. Thenon-transitory, computer-readable medium of claim 9, the set of actsfurther comprising: based on the determined operation status of theprinting device, transmitting a message indicating the status of theprinting device.
 16. The non-transitory, computer-readable medium ofclaim 9, wherein the printing device is a first printing device of aplurality of printing devices, and wherein determining the operationalstatus of the printing device based on the determined set ofcoefficients of the Markov chain further comprises: determining theoperational status of each of the plurality of printing devices based onthe determined set of coefficients of the Markov chain.
 17. A systemcomprising: one or more processors; and a memory storage havinginstructions stored thereon that when executed by the one or moreprocessors cause the system to perform a set of acts comprising:determining a plurality of time intervals for a printing deviceincluding a first time interval and a second time interval; determininga first number of pages printed by the printing device during the firsttime interval and a second number of pages printed by the printingdevice during the second time interval; determining a first number ofprinting-device errors that occur on the printing device during thefirst time interval and a second number of printing-device errors thatoccur on the printing device during the second time interval, whereineach printing-device error has a corresponding error type; calculating afirst ratio of pages printed per error, based on the first number ofpages and the first number of printing-device errors, and calculating asecond ratio of pages printed per error, based on the second number ofpages and the second number of printing-device errors; comparing thefirst ratio to one or more predetermined thresholds, and comparing thesecond ratio to the one or more predetermined thresholds; determining,based on the comparison of the first ratio with the one or morepredetermined thresholds, a first operational state of the printingdevice during the first time interval, wherein the first operationalstate is one of two or more operational states; determining, based onthe comparison of the second ratio with the one or more predeterminedthresholds, a second operational state of the printing device during thesecond time interval, wherein the second operational state is differentfrom the first operational state and is one of the two or moreoperational states; determining a set of one or more coefficients of aMarkov chain based on the first and second operational states, and oneor more transitions between the two or more operational states; based onthe determined set of coefficients, determining an operational status ofthe printing device.
 18. The system of claim 17, wherein the printingdevice has a counter that is incrementally increased for each pageprinted by the printing device, and wherein determining the number ofpages printed by the printing device during the first and second timeintervals comprises determining the number of pages based on thecounter.
 19. The system of claim 17, the set of acts further comprising:based on the determined operational status of the printing device,classifying the printing device as either healthy, unhealthy, orunknown.
 20. The system of claim 17, the set of acts further comprising:based on the determined operation status of the printing device,transmitting a message indicating the status of the printing device. 21.The method of claim 1, wherein the error type corresponding to eachprinting device error includes a high cost error and a low cost error,the method further comprising: determining a threshold value for thenumber of errors; and determining four or more operational states, thefour or more states including a first state corresponding to a statewith the high cost error and the number of errors below the threshold, asecond state corresponding to a state with the high cost error and thenumber of errors above the threshold, a third operational statecorresponding to a state with the low cost error and the number oferrors below the threshold, and a fourth operational state correspondingto a state with the low cost error and the number of errors above thethreshold, wherein a first coefficient of the set of coefficients isdetermined based on whether the error type is the high cost error or thelow cost error, and based on whether the first number of printing-deviceerrors is above or below the threshold value, and a second coefficientof the et of coefficients is determined based on whether the error typeis the high cost error or the low cost error, and based on whether thesecond number of printing-device errors is above or below the thresholdvalue, and wherein determining a first operational status comprisesdetermining the first operational status based on the first, second,third, and fourth operational states and the first coefficient, anddetermining a second operational status comprises determining the firstoperational status based on the first, second, third, and fourthoperational states and the second coefficient.
 22. The method of claim21, wherein the error type corresponding to each printing-device errorincludes a middle cost error.
 23. The method of claim 1, whereindetermining a transition between the first operational state and thesecond operational state includes determining whether the operationalstate transitions from one state to a different state, or determiningthat the operational state remains in the same operational state. 24.The method of claim 1, wherein determining the plurality of timeintervals includes determining the plurality of time intervals for twoor more printing devices, and wherein determining the set ofcoefficients includes determining a coefficient for a shared Markovchain.
 25. The method of claim 1, wherein determining the plurality oftime intervals includes determining the plurality of time intervals fortwo or more printing devices, wherein the two or more printing devicesare each associated with respective Markov Chains and coefficients.